[Paper Review] Deterministic and random features for large-scale quantum kernel machine

Kouhei Nakaji, Hiroyuki Tezuka+1 more|arXiv (Cornell University)|Sep 5, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes deterministic and random feature maps for large-scale quantum kernel machines, enabling scalable and generalizable quantum machine learning. By showing that quantum kernels with reduced density matrices (RDM) can be approximated via low-dimensional feature vectors with guaranteed error bounds, the method overcomes the scalability issue of quadratic classical cost while preserving generalization, validated through experiments on datasets with 1,000–10,000 samples.

ABSTRACT

Quantum machine learning (QML) is the spearhead of quantum computer applications. In particular, quantum neural networks (QNN) are actively studied as the method that works both in near-term quantum computers and fault-tolerant quantum computers. Recent studies have shown that supervised machine learning with QNN can be interpreted as the quantum kernel method (QKM), suggesting that enhancing the practicality of the QKM is the key to building near-term applications of QML. However, the QKM is also known to have two severe issues. One is that the QKM with the (inner-product based) quantum kernel defined in the original large Hilbert space does not generalize; namely, the model fails to find patterns of unseen data. The other one is that the classical computational cost of the QKM increases at least quadratically with the number of data, and therefore, QKM is not scalable with data size. This paper aims to provide algorithms free from both of these issues. That is, for a class of quantum kernels with generalization capability, we show that the QKM with those quantum kernels can be made scalable by using our proposed deterministic and random features. Our numerical experiment, using datasets including $O(1,000) \sim O(10,000)$ training data, supports the validity of our method.

Motivation & Objective

• Address the scalability and generalization limitations of quantum kernel methods (QKM) in near-term quantum machine learning.
• Identify and analyze quantum kernels with generalization capability (QKGC), focusing on those based on reduced density matrices (RDM).
• Develop feature-based approximations (DQF and RQF) that transform QKM into a linear problem with reduced computational cost.
• Demonstrate that RDM-based QKGCs can be efficiently approximated using feature maps, enabling large-scale applications.
• Provide theoretical guarantees on approximation error and computational complexity reduction for the proposed feature maps.

Proposed method

• Define quantum kernels with generalization capability (QKGC), focusing on RDM-based kernels as a viable class.
• Introduce deterministic quantum features (DQF) as a low-dimensional feature map that approximates the RDM-based quantum kernel via inner products.
• Propose randomized quantum features (RQF) as a stochastic variant of DQF, enabling efficient sampling with controlled error bounds.
• Theoretical analysis proves that both DQF and RQF approximate the original RDM kernel with a guaranteed error bound, ensuring generalization and scalability.
• Use the feature maps to convert kernel ridge regression into a linear model with O(MD²) complexity, where D is the feature dimension and M the number of data points.
• Validate the method using numerical experiments on the wine quality and magic04 datasets with 1,000–10,000 samples, comparing against the Nystrom method.

Experimental results

Research questions

• RQ1 Can quantum kernel methods with generalization capability be made scalable for large datasets?
• RQ2 Are RDM-based quantum kernels amenable to low-dimensional feature approximation with theoretical error guarantees?
• RQ3 How do deterministic and randomized quantum features (DQF and RQF) compare in performance and efficiency to classical kernel approximation methods like Nystrom?
• RQ4 Can feature-based representations of QKGCs serve as a design principle for new quantum kernel methods with quantum advantage?
• RQ5 What is the impact of tuning feature map parameters (e.g., embedding parameters) on the performance of the resulting machine learning models?

Key findings

• The proposed deterministic and randomized quantum features (DQF and RQF) successfully approximate RDM-based quantum kernels with a guaranteed error bound.
• The computational cost of the resulting machine learning model is reduced from O(M³) to O(MD²), enabling scalability to datasets with O(1,000) to O(10,000) samples.
• Numerical experiments show that DQF outperforms the Nystrom method in both regression and classification tasks on the wine quality and magic04 datasets.
• RQF does not outperform the Nystrom method on the same datasets, but offers flexibility for future tuning and optimization.
• Theoretical analysis confirms that quantum kernels based on reduced density matrices (RDM) are the primary viable class of QKGCs, as other constructions (e.g., reduced observable) are nearly equivalent or lack quantum advantage.
• The feature representation framework provides a new design guideline: any new QKGC should be expressible via feature vectors to ensure scalability and generalization.